Kakatiya University (KU) 2010-2nd Year B.Sc Mathematics B.A./ (ABSTRACT ALGEBRA

33. If R is a ring with unity and N is an ideal of R such that N? R then obtain the unity of R/N.

34. Let R be commutative ring and let aIR. Show that Ia = {xIR/ax = 0}is an ideal of R.

35. Let R be a ring with contains at lowest 2 elements. Suppose for every non zero a IR, there exists a unique b IR such that aba =a.

(a) Show that R has no devisors of zero.

(b) Show that bab = b

36. Let A and B are ideals of a ring R, the sum A+B of A and B is described by A+B={a+b/aIA, b I B}.

(a) Show that A+B is an ideal (b) Show that A K A+B and B K A+B.


UNIT-III

37. a) Use the definition of the limit of a sequence to establish



b) If 0 < a < b, determine



38. Determine the limits of



39. Let x1 = eight and xn+1 = ½ xn + two for n I N. Show that (xn) is bounded and monotone. obtain the limit.


40. Let x1 > one and xn+1 = two - 1/xn for n I N. Show that (xn) is bounded and monotone. obtain the limit.


41. Let x1 = one and xn+1 = for n I N. Show that (xn) converges and obtain the limit.


42. Let for every n I N. Prove that (xn) is increasing and bounded, and hence converges.


43. Establish the convergence and obtain the limits of the sequences


i) ii)

44. a) Establish the convergence and obtain the limit of

b) Determine the limit of


45. a) Establish the convergence and obtain the limit of

b) Determine the limit of


46. a) Show that the sequence is a Cauchy sequence.

b) Show that is not a Cauchy sequence.


47. If x1 < x2 are arbitrary real numbers and for n > 2, show that (xn) is convergent. What is its limit?





48. a) Show that .

b) Show that the series is convergent.


49. a) Show that

b) Show that the series is convergent.


50. Use the Cauchy condensation test to

a) Establish the divergence of

Earning:   Approval pending.